Subject: Re: two questions From: Jeffrey Strom Date: Thu, 22 Jun 2006 22:23:25 -0400 > > Subject: weak equivalence pi_k(X)-->pi_k(Y) if k>n > From: Gaucher Philippe > Date: Thu, 22 Jun 2006 16:00:01 +0200 > > Dear All > > Is there a known construction on the category of compactly generated > topological spaces, or on the category of simplicial sets of a model category > structure such that X-->Y is a weak equivalence iff pi_k(X)-->pi_k (Y) is an > isomorphism for k > n (n fixed). > > (for k S^n-->D^{n+1}). > > Thanks in advance. pg. Just a thought about this. I think Neisendorfer localization is relevant here (that is, nullification with respect to BZ/p followed by p-completion -- I'll denote it by L). If X is a 2-connected finite complex which is p-complete (i.e., finite, p-local homotopy groups), and X(n) is its n-connected cover, then L(X(n)) = X, whatever n is. Therefore, if f:X --> Y induces isomorphisms pi_k(X) --> pi_k(Y) for k > n, then X(n) --> Y(n) is a weak equivalence, and (assuming the Neisendorfer result is natural enough, which I'm pretty sure it is), then X = L(X(n)) --> L(Y(n)) = Y is also a weak equivalence. Jeff